What do physicists mean by time?
We’ll start with the easy question what do physicists mean by time.
Note that it’s easy to get mixed up between the concepts of time and the flow of time. When non-physicists talk about time they usually mean the flow of time i.e. the fact that in the human experience time flows inexorably onwards (at one second per second). We’ll get on to this, but for now we’ll ignore the question of why time flows and just address what time means to a physicist.
If you want to locate some position in space one method is to set up some axes, e.g. $x$, $y$ and $z$ axes, and you can then uniquely identify any point in space by its coordinates $(x, y, z)$.

To distinguish between events happening at the same point in space but at different times we need to specify when an event happened as well as where it happened, so we add a time coordinate $t$. Events can then be uniquely located by their spacetime coordinates $(t, x, y, z)$. To a physicist time is just a coordinate used to specify events in spacetime. In figure 1 above we have an $x$ axis stretching from $-\infty$ to $\infty$, a $y$ axis stretching from $-\infty$ to $\infty$ and a $z$ axis stretching from $-\infty$ to $\infty$. To these a physicist adds a $t$ axis stretching from $-\infty$ to $\infty$, and that’s what time is - just a coordinate.
But everyday experience tells us that time is special - certainly different from space - so what justifies the physicist’s view that time is just a coordinate? To understand this start with time in the everyday world as described by Newtonian mechanics.
Suppose I set up a coordinate system with myself at the origin, $x$ to the East, $y$ to the North and $z$ straight up. For time I’ll use my wristwatch. And suppose you do the same, but let’s say you’re in a different country from me. Our two sets of coordinates won’t match, because our East, North and up axes point in different directions.

Or suppose you are moving relative to me. Even ignoring the curvature of the Earth’s surface, our coordinates won’t match because your origin is constantly moving relative to my origin - what appears to be stationary to me is moving in your coordinates and vice versa.
So spatial coordinates are observer dependent. However time is absolute. Assuming we both use Greenwich Mean Time (or some other similar standard) we will always both agree on the time no matter where we are on Earth or however we are moving relative to each other. In Newtonian mechanics time is special for this reason, so it makes sense to consider it separately from space.
However since 1905 we have known that to properly describe the world around us we have to use special relativity, and in relativity time is not the same for all observers. Let’s go back to ordinary Newtonian mechanics for a moment, and suppose you’re moving relative to me along the $x$ axis at some speed $v$. If we draw my time $t$ and position $x$ axes and your $t’$ and $x’$ axes they’d look like:

Our two time axes point in the same direction, so we both agree on what it means to define a time axis. But now suppose you’re moving at relativistic speed $v$ and draw the same diagram.

When we include special relativity our axes no longer point in the same direction. If I draw my time axis straight up then relative to me your time axis is rotated by an angle $\theta$ given by:
$$ \tan(\theta) = \frac{v}{c} $$
So your time direction is a mixture of my time and space directions. You would see exactly the same - if you draw your time axis straight up then you’d see my time axis rotated by $-\theta$. In effect we have different definitions of time, and indeed this is why we get time dilation in relativity.
The point of all this is that in relativity time is not uniquely defined. When we consider the coordinates used by different observers we find that time and space get mixed up with each other. Time is no longer distinct from space, and that’s why physicists treat it as just one of the four coordinates that together make up four dimensional spacetime.
How does time flow?
The previous section explained what physicists mean by time, but made no mention of time flowing. This is because in relativity time doesn’t flow - more precisely the flow of time doesn’t exist as a concept.
This is going to take some explaining, so let me attempt it using a simple example. Suppose I throw you a ball and you catch it. Everyday experience tells us that time flows forwards and as it does so the ball rises up from my hand then falls down to your hand. If we graph the height of the ball, $h$, against time, $t$, we’ll get something like:

In Newtonian physics this has a nice simple interpretation: time flows forwards and the height is a function of time. We can write the height as $h(t)$. But now let me draw a different graph. I’ll graph the height of the ball, $h$, against the distance, $d$, the ball travels horizontally as it moves from me to you:

This looks awfully like the previous graph, and indeed I can write the height of the ball as a function of the horizontal distance travelled, $h(d)$. But we wouldn’t say that distance $d$ flows forward and the height changes as it does so, because, well, time is different from distance. The two graphs are just different views of a four dimensional graph showing the trajectory of the ball in spacetime (I’m only going to draw three dimensions because I can’t do 4D graphs):

In the previous section I went to some lengths to explain that time is just a coordinate, like the spatial coordinates, so this graph doesn’t show time flowing any more than distance or height are flowing. The trajectory of the ball is just a line in a 4D.
In relativity we call graphs like the above world lines, where the world line is just the set of all spacetime points $(t, x, y, z)$ that the ball occupies during its trajectory. This world line is a fixed object in four dimensional spacetime - it doesn’t change with time. All that changes is the ball’s position on the world line. This is why we say that time doesn’t flow. Time is just one of the four dimensions that the world line occupies.
In fact any physical property, pressure of a gas, strength of a gravitational field, or whatever, can be written as a function in the four spacetime dimensions, $F(t, x, y, z)$. Written this way the geometrical object $F$ exists in all of space and all of time - it’s not something that evolves in time any more than it’s something that evolves in space. In principle we could have some function that represented the whole universe, $\mathcal{F}(t, x, y, z)$, and this would exist for all values of $t$, $x$, $y$ and $z$. This idea (or a range of ideas like it) is called the block universe - the idea that the whole universe exists simultaneously and time doesn’t flow.
At this point I should note that many physicists, and I would guess the vast majority of non-physicists, would say this is just mathematical skulduggery and it’s nonsense to say time doesn’t flow. I’m not going to make any comment, except to say that this nicely brings us onto the last of our questions.
Why is there an arrow of time?
However mathematically convincing the idea of a block universe may be, the fact remains that our everyday experience tells us that:
time flows
it flows in one direction — forwards, and never backwards
So, how do we reconcile this with the idea of a block universe? Many physicists have expended much thought on this, and there are lots of differing views. However there is a something of a consensus that it is related to entropy. Indeed this is encapsulated in the second law of thermodynamics, which roughly speaking states that for any isolated system entropy only ever increases.
Consider some mechanism. We won’t worry exactly what it is, for example it could be something mechanical, an interstellar gas cloud or a human brain. When we talk about time flowing forward we mean that the state of the machine changes in a specific direction, e.g. a clock ticks forwards, and the second law of thermodynamics tells us that it changes in the direction of increasing entropy.
Assuming the human brain is just a mechanism, it changes in the direction of increasing entropy just like every other mechanism. But if consciousness is the result of the brain changing then it follows that any conscious being will observe mechanisms changing in the direction of increasing entropy. This isn’t so much a physical law as a correlation. Since our brains change in the same direction (of increasing entropy) as everything else that means they will necessarily observe everything to be changing in this same direction. We call this direction increasing time.
If I’m allowed a personal opinion I would say this all seems a little trite — too good to be true — and it seems a suspiciously simple explanation for something as complicated as the universe. However I have no better suggestion to make. Indeed, I don’t think anyone has a better suggestion, or at least not one better enough to convince large swathes of the physics community.